Subgraph and homomorphism counting are fundamental algorithmic problems. Given a constant-sized pattern graph $H$ and a large input graph $G$, we wish to count the number of $H$-homomorphisms/subgraphs in $G$. Given the massive sizes of real-world graphs and the practical importance of counting problems, we focus on when (near) linear time algorithms are possible. The seminal work of Chiba-Nishizeki (SICOMP 1985) shows that for bounded degeneracy graphs $G$, clique and $4$-cycle counting can be done linear time. Recent works (Bera et al, SODA 2021, JACM 2022) show a dichotomy theorem characterizing the patterns $H$ for which $H$-homomorphism counting is possible in linear time, for bounded degeneracy inputs $G$. At the other end, Ne\v{s}et\v{r}il and Ossona de Mendez used their deep theory of "sparsity" to define bounded expansion graphs. They prove that, for all $H$, $H$-homomorphism counting can be done in linear time for bounded expansion inputs. What lies between? For a specific $H$, can we characterize input classes where $H$-homomorphism counting is possible in linear time? We discover a hierarchy of dichotomy theorems that precisely answer the above questions. We show the existence of an infinite sequence of graph classes $\mathcal{G}_0$ $\supseteq$ $\mathcal{G}_1$ $\supseteq$ ... $\supseteq$ $\mathcal{G}_\infty$ where $\mathcal{G}_0$ is the class of bounded degeneracy graphs, and $\mathcal{G}_\infty$ is the class of bounded expansion graphs. Fix any constant sized pattern graph $H$. Let $LICL(H)$ denote the length of the longest induced cycle in $H$. We prove the following. If $LICL(H) < 3(r+2)$, then $H$-homomorphisms can be counted in linear time for inputs in $\mathcal{G}_r$. If $LICL(H) \geq 3(r+2)$, then $H$-homomorphism counting on inputs from $\mathcal{G}_r$ takes $\Omega(m^{1+\gamma})$ time. We prove similar dichotomy theorems for subgraph counting.

翻译：子图与同态计数是基础的算法问题。给定一个规模恒定的模式图$H$和一个大型输入图$G$，我们希望统计$G$中$H$-同态/子图的数量。鉴于现实世界图数据的庞大体积以及计数问题的实际重要性，我们聚焦于何时能够实现（近乎）线性时间算法。Chiba与Nishizeki的开创性工作（SICOMP 1985）表明，对于有界退化图$G$，团和4-环计数可以在线性时间内完成。近期研究（Bera等人，SODA 2021，JACM 2022）提出了一个二分定理，刻画了在输入$G$具有有界退化性时，可实现线性时间$H$-同态计数的模式$H$。另一方面，Nešetřil与Ossona de Mendez利用其深刻的“稀疏性”理论定义了有界扩张图。他们证明，对于所有$H$，在有界扩张输入的条件下，$H$-同态计数均可在线性时间内完成。那么介于两者之间的情况如何？对于特定$H$，我们能否刻画使得$H$-同态计数可在线性时间内完成的输入类别？我们发现了一系列二分定理构成的层次结构，精确回答了上述问题。我们证明了存在无限序列的图类$\mathcal{G}_0$ $\supseteq$ $\mathcal{G}_1$ $\supseteq$ ... $\supseteq$ $\mathcal{G}_\infty$，其中$\mathcal{G}_0$为有界退化图类，$\mathcal{G}_\infty$为有界扩张图类。固定任意规模恒定的模式图$H$，设$LICL(H)$表示$H$中最长导出环的长度。我们证明如下结论：若$LICL(H) < 3(r+2)$，则对于$\mathcal{G}_r$类中的输入，$H$-同态可在线性时间内计数；若$LICL(H) \geq 3(r+2)$，则对于$\mathcal{G}_r$类中的输入，$H$-同态计数需要$\Omega(m^{1+\gamma})$时间。我们为子图计数证明了类似的二分定理。